This invention relates to the measurement of high-order derivatives of device parameters and, more particularly, to a frequency mixing scheme for measuring d.sup.n V/dI.sup.n (n.gtoreq.2) of nonlinear devices especially those having an exponential relationship between voltage V and current I.
Derivative techniques have been widely used to detect fine structure in the electrical characteristics of semiconductor devices such as tunnel diodes (Korb et al., Rev. Sci. Instr., Vol. 43 p. 90 (1972)), transistors, junction diodes, and most recently double-heterostructure junction lasers. The experimental technique commonly involves sinusoidally modulating an independent variable, such as current or voltage, and detecting the ac response of the dependent variable. A measure of the first derivative of the electrical characteristic is then provided by the ac component of the device response at the frequency of modulation, while the second derivative is in principle proportional to the sinusoidal response at twice the modulation frequency. However, as pointed out by Korb et al. supra, a practical realization of the second derivative measurement encounters two problems. First, the detection of the second harmonic signal requires rejection of the fundamental signal, which for weakly nonlinear characteristics can be one or more orders of magnitude greater than the harmonic signal. Adequate attenuation of the fundamental without appreciable loss for the second harmonic requires a frequency selectivity which is not easily attained without sophisticated and expensive electronics. Without adequate rejection, the strong fundamental may saturate the low signal preamplification required to detect the weak second harmonic or at least contribute spuriously to its magnitude. Secondly, the presence of even a small degree of harmonic distortion in the modulation signal produces a second harmonic signal proportional to the first derivative and consequently an incorrect measure of the second derivative.